Differential Entropy - Differential Entropies For Various Distributions

Differential Entropies For Various Distributions

In the table below is the gamma function, is the digamma function, is the beta function, and γE is Euler's constant. Each distribution maximizes the entropy for a particular set of functional constraints listed in the fourth column, and the constraint that x be included in the support of the probability density, which is listed in the fifth column.

Table of differential entropies and corresponding maximum entropy constraints
Distribution Name Probability density function (pdf) Entropy in nats Maximum Entropy Constraint Support
Uniform None
Normal
Exponential
Rayleigh
Beta for

Cauchy
Chi
Chi-squared
Erlang
F
Gamma
Laplace
Logistic
Lognormal
Maxwell-Boltzmann
Generalized normal
Pareto
Student's t
Triangular f(x) = \begin{cases} \frac{2(x-a)}{(b-a)(c-a)} & \mathrm{for\ } a \le x \leq c, \\ \frac{2(b-x)}{(b-a)(b-c)} & \mathrm{for\ } c < x \le b, \\ \end{cases}
Weibull
Multivariate normal f_X(\vec{x}) =

(Many of the differential entropies are from.

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